essential matrix

The essential matrix is the calibrated sibling of the fundamental matrix. When you already know each camera's internal parameters (focal length, principal point), you can strip them away and work with rays instead of pixels — directions in 3D measured from the camera center. In these normalized coordinates the epipolar constraint becomes x' transposed times E times x equals zero, where now x and x' are calibrated ray directions, not raw pixels. The link between the two matrices is exact: E equals the second camera's intrinsic matrix transposed, times F, times the first camera's intrinsic matrix.

What makes E special is that it cleanly factors into the two things you actually want: the relative rotation and translation between the cameras. Specifically E equals the cross-product matrix of the translation vector, times the rotation matrix — written compactly as E = [t]× R. The cross-product matrix [t]× is the 3×3 matrix that performs 'cross with t' as a multiplication. Because of this structure, E has only 5 degrees of freedom (3 for rotation, 2 for translation direction — the length of the translation is unrecoverable from images alone, which is the famous global scale ambiguity).

To turn E into pose you take its singular value decomposition and read off the rotation and the translation direction. The algebra yields four candidate solutions — two possible rotations paired with two possible translation signs — that all satisfy the epipolar equation but correspond to different physical configurations (cameras facing each other, mirrored, and so on). You disambiguate with the cheirality check: triangulate a few matched points under each candidate and keep the one solution that places those points in front of both cameras rather than behind. The minimal estimator is Nister's 5-point algorithm, which recovers E from just five correspondences and is the workhorse inside RANSAC for structure-from-motion and visual SLAM front ends.

Conceptually, the fundamental matrix says 'these two pixels could be a match,' while the essential matrix says 'and here is the actual rotation and translation direction that makes them a match.' E is the bridge from raw image geometry to a metric (up-to-scale) reconstruction of how the camera moved.